### Solving Systems of Equations by Graphing Worksheets

A system of linear equations is composed of two or more equations defined simultaneously. A system of equations in two variables is consists of two equations and two unknowns. The solution to system of equations is a point that satisfies both equations. If graphs are given, the solution set is the point of intersection. If the two graphs intersect at a point then it has one solution. If the lines coincide, then the solution is the set of all points on the line, so the system has infinitely many solutions. If the two lines do not intersect, then there is no real solution. These solutions can be found either by graphing or by algebraic operations. Unless the graphs of the system are graphed accurately, then graphical method is appropriate. To check if your answer is reasonable, you may try to use any of the algebraic methods.

To solve a system by graphing, use the following steps:

1. If the equations are given, sketch the lines corresponding to these equations in the same coordinate axes. Use previously discussed methods of graphing linear equations, whichever method is most appropriate to the given system.
2. Locate the point of intersection. You can proceed to this step, if the graphs are already given.
3. Check your solution by substituting the ordered pair to both equations. State the solution set.
Examples

Solve each system by graphing.
$\small 1. \;\;\displaystyle \left\{\begin{matrix} x-3y=6\\ x+2y=-9 \end{matrix}\right$
Solution:

Graph each line using slope-intercept method.
Rewrite each equation into slope-intercept form.

\small \begin{align*} & -3y=-x+6 &\textup{Rewriting the first equation}\\ & y=\displaystyle \frac{1}{3}x-2&\textup{Dividing both sides by -3}\\ \\&2y=-x-9&\textup{Rewriting the second equation}\\&y=-\displaystyle \frac{1}{2}x-4\displaystyle \frac{1}{2} &\textup{Dividing both sides by 2}\end{align*}

Figure 1 below shows the graph of the system:
$\small \displaystyle \left\{\begin{matrix} x-3y=6\\ x+2y=-9 \end{matrix}\right$

Figure 1

As noted, the two lines intersect at $$\left(-3,-3 \right)$$ so the solution to this system is the point $$\left(-3,-3 \right)$$.

$\small 2. \;\;\displaystyle \left\{\begin{matrix} x+2y=3\\ 2x-y=1 \end{matrix}\right$

Solution:

Graph each line using the intercept method. Find both the $$x-$$ and $$y-$$ intercepts of each line. Then sketch the graphs.

The $$x-$$ and $$y-$$ intercepts of $$x+2y=3$$ are 3 and 1.5, respectively. The $$x-$$ and $$y-$$ intercepts of $$2x-y=1$$ are $$\frac{1}{2}$$ and $$–1$$ , respectively.

Figure 2 below shows the graph of this system of equations.
Figure 2

It can be seen that the two lines intersect at $$\left(1,1 \right)$$. The solution to this system is $$\left(1,1 \right)$$.

Practice Exercises

Solve each system below by graphing.
$\small 1. \;\;\displaystyle \left\{\begin{matrix} x+2y=3\\ 2x-y=1 \end{matrix}\right$
$\small 2. \;\;\displaystyle \left\{\begin{matrix} x-2y=-2\\ 2x-y=2 \end{matrix}\right$
$\small 3. \;\;\displaystyle \left\{\begin{matrix} 3x=2y+2\\ 2x-y=4 \end{matrix}\right$